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A063037 Numbers without 3 consecutive equal binary digits. 10
0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 25, 26, 27, 36, 37, 38, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 73, 74, 75, 76, 77, 82, 83, 84, 85, 86, 89, 90, 91, 100, 101, 102, 105, 106, 107, 108, 109, 146, 147, 148, 149, 150, 153, 154, 155, 164, 165, 166 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Complement of A136037; intersection of A003796 and A003726. - Reinhard Zumkeller, Dec 11 2007

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

It appears (but has not yet been proved) that the terms of {a(n)} can be computed recursively as follows. Let {c(i)} be defined for i >= 4 by c(i) = 2c(i-1) + 1, if n is a multiple of 3, else c(i) = 2c(i-1) - 1, with c(4) = 1. I.e., {c(i)} = {1,1,3,5,9,19,37,73,147,...}, for i=4,5,6,... . Let a(1)=1, a(2)=2, a(3)=3. For n > 3, choose k so that F(k)-2 < n <= F(k+1)-2, where F(k) denotes the k-th Fibonacci number (A000045). Then a(n) = c(k) + 2a(F(k)-2) - a(2F(k)-n-3). This has been verified for n up to 1100. - John W. Layman, May 26 2009

EXAMPLE

The binary representation of 9 (1001) has no 3 consecutive equal digits.

MAPLE

isA063037 := proc(n)

    local bdgs, rep, d, i ;

    if n = 0 then

        return true;

    end if;

    bdgs := convert(n, base, 2) ;

    rep := 1;

    d := op(1, bdgs) ;

    for i from 2 to nops(bdgs) do

        if op(i, bdgs) = op(i-1, bdgs) then

            rep := rep+1 ;

        else

            rep :=1 ;

        end if ;

        if rep > 2 then

            return false;

        end if;

    end do:

    return true ;

end proc:

for n from 0 to 50 do

    if isA063037(n) then

        printf("%d, ", n) ;

    end if;

end do: # R. J. Mathar, Dec 18 2013

MATHEMATICA

Select[Range[0, 168], AllTrue[Length /@ Split@ IntegerDigits[#, 2], # < 3 &] &] (* Michael De Vlieger, Aug 20 2017 *)

PROG

(PARI) { n=0; for (m=0, 10^9, x=m; t=1; b=2; while (x>0, d=x-2*(x\2); x\=2; if (d==b, c++; if (c==3, t=x=0), b=d; c=1)); if (t, write("b063037.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 16 2009

CROSSREFS

Cf. A000045, A000975, A286262.

Sequence in context: A032878 A032845 A023776 * A286262 A201992 A236562

Adjacent sequences:  A063034 A063035 A063036 * A063038 A063039 A063040

KEYWORD

easy,nonn

AUTHOR

Lior Manor, Jul 05 2001

EXTENSIONS

Missing "less than" sign supplied in the conjectured recurrence (thanks to Franklin T. Adams-Watters for pointing this out) by John W. Layman, Nov 09 2009

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.